Related papers: How to Compute a Puiseux Expansion
In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of…
We obtain an asymptotic expansion for $p(n)$, the number of partitions of a natural number $n$, starting from a formula that relates its generating function $f(t), t\in (0,1)$ with the characteristic functions of a family of sums of…
In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
In this paper, we present fast algorithms for the product of two multivariate polynomials in sparse representation. The bit complexity of our algorithms are studied in detail for various types of coefficients, and we derive new complexity…
We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
We present a simple and fast algorithm for the computation of the coefficients of the expansion of a function f(cos u) in ultraspherical (Gegenbauer) polynomials. We prove that these coefficients coincide with the Fourier coefficients of an…
We propose a method to compute the numerical solutions of a polynomial system in complete intersection. This algorithm makes use of Bezout matrices and need only linear algebra computations. All the calculations can be done in floating…
A new method of numerical solution for partial differential equations is proposed. The method is based on a fast matrix multiplication algorithm. Two-dimensional Poison equation is used for comparison of the proposed method with…
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions.…
We show that the roots of a Weierstrass polynomial on several variables are power series with rational exponents of bounded denominators, whose Newton diagrams are contained in an S-cone. As an application, in Section 4 we deal with some…
This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of…
We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…
We use the periodicity properties of generalized Gauss sums to factor numbers. Moreover, we derive rules for finding the factors and illustrate this factorization scheme for various examples. This algorithm relies solely on interference and…
Pretropisms are candidates for the leading exponents of Puiseux series that represent solutions of polynomial systems. To find pretropisms, we propose an exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton…
In this paper we use Euler-Seidel matrices method to find out some properties of exponential and geometric polynomials and numbers. Some known results are reproved and some new results are obtained.
We show that explicit forms for certain polynomials~$\psi^{(a)}_m(n)$ with the property \[ \psi^{(a+1)}_m(n) = \sum_{\nu=1}^n \psi_m^{(a)}(\nu) \] can be found (here, $a,m,n\in\mathbb{N}_0$). We use these polynomials as a basis to express…
A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of…